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Mathematical modeling of the onset of capillary formation initiating angiogenesis. (English) Zbl 0977.92013

Summary: It is well accepted that neo-vascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel. We present a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In this model, a single layer of endothelial cells is separated by a vascular wall from an extracellular tissue matrix. A coupled system of ordinary and partial differential equations is derived which, in the presence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the extracellular matrix. We refer to this as the onset of vascular sprouting. Some biological evidence for the correctness of our model is indicated by the formation of teats in utero. Further evidence for the correctness of the model is given by its prediction that endothelial cells will line the nascent capillary at the onset of capillary angiogenesis.

MSC:

92C45 Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
60G50 Sums of independent random variables; random walks
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C37 Cell biology
92C50 Medical applications (general)
92C15 Developmental biology, pattern formation
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